DOCSLIB.ORG

On Analytic Formulas of Feynman Propagators in Position Space *

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Analytic Formulas of Feynman Propagators in Position Space * CPC(HEP & NP), 2010, 34(10): 1576{1582 Chinese Physics C Vol. 34, No. 10, Oct., 2010 On analytic formulas of Feynman propagators in position space * ZHANG Hong-Hao(Ü÷Ó)1;1) FENG Kai-Xi(¾mU)1 QIU Si-Wei(£d)1 ZHAO An(ëS)1 LI Xue-Song(oÈt)2 1 School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China 2 Science College, Hunan Agricultural University, Changsha 410128, China Abstract We correct an inaccurate result of previous work on the Feynman propagator in position space of a free Dirac field in (3 + 1)-dimensional spacetime; we derive the generalized analytic formulas of both the scalar Feynman propagator and the spinor Feynman propagator in position space in arbitrary (D + 1)- dimensional spacetime; and we further find a recurrence relation among the spinor Feynman propagator in (D+1)-dimensional spacetime and the scalar Feynman propagators in (D+1)-, (D−1)- and (D+3)-dimensional spacetimes. Key words Feynman propagator, Klein-Gordon field, Dirac field, arbitrary dimensional spacetime PACS 11.10.Kk, 03.70.+k, 02.30.Gp 1 Introduction language of quantum field theory, the Feynman prop- agator in position space can be physically understood Although one cannot adopt the extreme view that as the energy due to the presence of the two external the set of all Feynman rules represents the full the- sources located at two different points in space and ory of quantized fields, the approach of the Feynman acting on each other (See Ref. [1] Chapt.I.4). graphs and rules plays an important role in pertur- It has been mentioned in Refs. [1{8] that in (3+1)- bative quantum field theories. For a generic quan- dimensional spacetime, after integrating over momen- tum theory involving interacting fields, the set of its tum, the Feynman propagator of a free Klein-Gordon Feynman rules includes some vertices and propaga- field can be expressed in terms of Bessel or modified tors. While for a free field theory there is only one Bessel functions, which depends on whether the sep- graph, that is, the Feynman propagator, in the set aration of two spacetime points is timelike or space- of its Feynman rules. Thus the Feynman propagator like. By changing variables to hyperbolic functions describes most, if not all, of the physical contents of and using the integral representation of the Hankel the free field theory. It is true that the real world is function of the second kind, the authors of Ref. [2] not governed by any free field theory. However, this derived the full analytic formulas of the Feynman kind of theory is the basis of a perturbatively inter- propagors of free Klein-Gordon and Dirac fields in acting field theory, and the issues of a free theory are (3+1)-dimensional spacetime. The expressions of the usually in the simplest situation and thus their study Feynman propagators of a free Klein-Gordon field in will be attempted at the first step of investigations. (1+1)- and that in (2+1)-dimensional spacetime can In statistical field theory, the Feynman propagator is be found in Refs. [9] and [10], respectively. How- usually called the correlation function. We need to ever, in Ref. [2] the expression for the Feynman prop- know its formula in position space to figure out the agator of a Dirac spinor field is inaccurate, since critical exponent η and the correlation length ξ, which there is at least a redundant term in their results. is related to another exponent ν. In the path integral In this paper we will show that the term actually Received 20 November 2009, Revised 21 January 2010 * Supported by Specialized Research Fund for Doctoral Program of Higher Education (SRFDP) (200805581030) and Sun Yet- Sen University Science Foundation and Fundamental Research Funds for the Central Universities 1) E-mail: [email protected] ©2010 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd No. 10 ZHANG Hong-Hao et al: On analytic formulas of Feynman propagators in position space 1577 vanishes and we will give the correct expression. Fur- Thus, we can obtain the analytic formula of the Feyn- thermore, we will generalize the results of previous man propagator in position space by integrating over work and derive the full analytic formulas of the Feyn- the D-dimensional momentum in the expression of man propagators in position space of, respectively, D( t ;~x). This integral in D-dimensional Euclidean j j the Klein-Gordon scalar and the Dirac spinor in arbi- space can be evaluated by changing the variables from trary (D + 1)-dimensional spacetime. Eventually we Cartesian coordinates to spherical coordinates. Since will find an interesting recurrence relation between the angular integral parts look a little different in the spinor Feynman propagator in (D+1)-dimensional (1+1)-, (2+1)- and general (D +1)-dimensional (for spacetime and the scalar Feynman propagators in D > 3) spacetimes, in order to be more careful in our (D + 1)- and alternate-successive dimensional space- derivation, let us consider these situations case by time. case. Eventually we will show that the general result This paper is organized as follows. In Section 2, of (D+1)-dimensional spacetime holds for D = 1;2 as we will briefly review the derivation of the analytic well. formulas of the Feynman propagators of a free Klein- 2.1 (1+1)-dimensional spacetime Gordon field in (1+1)- and (2+1)-dimensional space- time, and we will compute, once and for all, the When the spatial dimension D = 1, Eq. (3) be- scalar Feynman propagator in (D + 1)-dimensional comes +1 spacetime. After completion of this work, we became dp 1 −i(Ejt|−pr) DF(t;r) = π e ; (4) aware of the analytic formula of the scalar Feynman Z−∞ 2 2E propagator in position space in arbitrary dimensional with E = pp2 +m2. Using the substitution E = m spacetime which is also given in Ref. [11]. We will cosh η, p = m sinh η (with < η < ) in the above demonstrate that our result exactly agrees with that −∞ 1 integral, we have of Ref. [11] in this section. In Section 3 we will make 1 +1 use of the obtained formula of the scalar Feynman D (t;r) = dηe−im(jtj cosh η−r sinhη): (5) F π Z propagator to compute the expression of the spinor 4 −∞ Feynman propagator in (D+1)-dimensional spacetime Due to the Lorentz invariance, the Feynman propa- and obtain a recurrence relation. We will also com- gator can depend only on the interval x2 t2 r2. ≡ − pare our result for D = 3 with that of Ref. [2], and we If the interval is timelike, x2 > 0, we can make a will show that one additional term in Ref. [2] actually Lorentz transformation such that x is purely in the has no contribution and that the method they used time-direction, x0 = θ(t)pt2 r2 i, r = 0. Note − − to prove the term nonzero was inappropriate. The that ps is not a single-valued-function of s and here last section is devoted to conclusions. and henceforth the cut line in the complex plane of s is chosen to be the negative real axis. The nega- tive infinitesimal imaginary part, i, is because of 2 Feynman propagator of Klein- − the Feynman description of the Wick rotation, i.e., Gordon theory x2 x2 i in position space and correspondingly ! − k2 k2 +i in momentum space. Thus, ! Following the notation of Refs. [1, 4], we write +1 1 −impt2−r2−i cosh η the Feynman propagator of a free Klein-Gordon field DF(t;r) = π dηe 4 Z−∞ φ(x) φ(t;~x) in (D+1)-dimensional spacetime as the i ≡ = H (2)(mpt2 r2 i); (6) time-ordered two-point correlation function, −4 0 − − (2) DF(x) 0 T φ(x)φ(0) 0 where H0 (x) is the Hankel function of the second ≡ h j j i kind, and where we have used the identity in # 3.337 = θ(x0)D(x)+θ( x0 )D( x) ; (1) − − of Ref. [12], +1 with the unordered two-point correlation function, −iβ cosh η π (2) π dηe = i H0 (β); ( < argβ < 0): (7) Z−∞ − − dDp~ 1 −i(Ep~t−p~·~x) 2 D(x) 0 φ(x)φ(0) 0 = D e : (2) Likewise, if the interval is spacelike, x < 0, we have ≡ h j j i Z (2π) 2Ep~ +1 1 impr2−t2+i sinhη Combining the above two equations gives DF(t;r) = π dηe 4 Z−∞ dDp~ 1 −i(Ep~jt|−p~·~x) 1 D (t;~x) = D( t ;~x) = e : (3) 2 2 F π D = K0(mpr t +i); (8) j j Z (2 ) 2Ep~ 2π − 1578 Chinese Physics C (HEP & NP) Vol. 34 1 2 where K0(x) is the modified Bessel function, and + θ( x2) e−mp−x +i: (16) − 4πp x2 +i where we have used the identity in # 3.714 of − Ref. [12], Eqs. (13) and (16) agree well with the results of pre- 1 π π vious work [9, 10]. dη cos(β sinhη) = K0(β); < argβ < : (9) Z0 − 2 2 2.3 (D+1)-dimensional spacetime (for D > 3) It is worthwhile noting that the i description as- − sures the proper phase angles of px2 i in Eq. (6) Now, let us proceed to compute the scalar Feyn- − and p x2 +i in Eq. (8), respectively, so that the man propagator in (D+1)-dimensional spacetime (for − mathematical identities (7) and (9) can be applied D > 3).
Load more

Recommended publications
  • Cylindrical Waves Guided Waves
    Cylindrical Waves Guided Waves Cylindrical Waves Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648—Waves in Cylindrical Coordinates D. S. Weile Cylindrical Waves 2 Guided Waves Cylindrical Waveguides Radial Waveguides Cavities Cylindrical Waves Guided Waves Outline 1 Cylindrical Waves Separation of Variables Bessel Functions TEz and TMz Modes D. S. Weile Cylindrical Waves Cylindrical Waves Guided Waves Outline 1 Cylindrical Waves Separation of Variables Bessel Functions TEz and TMz Modes 2 Guided Waves Cylindrical Waveguides Radial Waveguides Cavities D. S. Weile Cylindrical Waves Separation of Variables Cylindrical Waves Bessel Functions Guided Waves TEz and TMz Modes Outline 1 Cylindrical Waves Separation of Variables Bessel Functions TEz and TMz Modes 2 Guided Waves Cylindrical Waveguides Radial Waveguides Cavities D. S. Weile Cylindrical Waves Separation of Variables Cylindrical Waves Bessel Functions Guided Waves TEz and TMz Modes The Scalar Helmholtz Equation Just as in Cartesian coordinates, Maxwell’s equations in cylindrical coordinates will give rise to a scalar Helmholtz Equation. We study it first. r2 + k 2 = 0 In cylindrical coordinates, this becomes 1 @ @ 1 @2 @2 ρ + + + k 2 = 0 ρ @ρ @ρ ρ2 @φ2 @z2 We will solve this by separating variables: = R(ρ)Φ(φ)Z(z) D. S. Weile Cylindrical Waves Separation of Variables Cylindrical Waves Bessel Functions Guided Waves TEz and TMz Modes Separation of Variables Substituting and dividing by , we find 1 d dR 1 d2Φ 1 d2Z ρ + + + k 2 = 0 ρR dρ dρ ρ2Φ dφ2 Z dz2 The third term is independent of φ and ρ, so it must be constant: 1 d2Z = −k 2 Z dz2 z This leaves 1 d dR 1 d2Φ ρ + + k 2 − k 2 = 0 ρR dρ dρ ρ2Φ dφ2 z D.
    [Show full text]
  • Appendix a FOURIER TRANSFORM
    Appendix A FOURIER TRANSFORM This appendix gives the definition of the one-, two-, and three-dimensional Fourier transforms as well as their properties. A.l One-Dimensional Fourier Transform If we have a one-dimensional (1-D) function fix), its Fourier transform F(m) is de­ fined as F(m) = [ f(x)exp( -2mmx)dx, (A.l.l) where m is a variable in the Fourier space. Usually it is termed the frequency in the Fou­ rier domain. If xis a variable in the spatial domain, miscalled the spatial frequency. If x represents time, then m is the temporal frequency which denotes, for example, the colour of light in optics or the tone of sound in acoustics. In this book, we call Eq. (A.l.l) the inverse Fourier transform because a minus sign appears in the exponent. Eq. (A.l.l) can be symbolically expressed as F(m) =,r 1{Jtx)}. (A.l.2) where ~-~denotes the inverse Fourier transform in Eq. (A.l.l). Therefore, the direct Fourier transform in this case is given by f(x) = r~ F(m)exp(2mmx)dm, (A.1.3) which can be symbolically rewritten as fix)= ~{F(m)}. (A.l.4) Substituting Eq. (A.l.2) into Eq. (A.l.4) results in 200 Appendix A ftx) = 1"1"-'{.ftx)}. (A.l.5) Therefore we have the following unity relation: ,.,... = l. (A.l.6) It means that performing a direct Fourier transform and an inverse Fourier transform of a function.ftx) successively leads to no change. Using the identity exp(ix) = cosx + isinx and Eq.
    [Show full text]
  • The Laplace Transform of the Modified Bessel Function K( T± M X) Where
    THE LAPLACE TRANSFORM OF THE MODIFIED BESSEL FUNCTION K(t±»>x) WHERE m-1, 2, 3, ..., n by F. M. RAGAB (Received in revised form 13th August 1963) 1. Introduction In the present paper we determine the Laplace transforms of the modified ±m Bessel function of the second kind Kn(t x), where m is any positive integer. The Laplace transforms are given in (2) and (4) below, p being the transform parameter and having positive real part. The formulae to be established are as follows (l)-(4): f Jo 2m-1 2\-~ -i-,lx 2m 2m 2m J k + v - 2 2m (2m)2mx2 v+1 v + 2 v + 2m (1) 2m 2m 2m where R(k±nm)>0 and x is taken for simplicity to be real and positive" When m = 1, x may be taken to be complex with real part greater than 1. The asterisk in the generalised hypergeometric function denotes that the . 2m . ., v+1 v + 2 v + 2m . factor =— m th e parametert s , , ..., is omitted; m is a positive 2m 2m 2m 2m integer. m m 1 e~"'Kn(t x)dt = 2 ~ V f V J ±l - 2m-l 2 2m 2m 2m 2m J 2m v+1 n v+1. 4p ' 2 2m 2 1m"' (2m)2mx2 2r2m+l v+1 v+2 v + 2m (2) _ 2m 2m ~lm~ E.M.S.—Y Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 05:34:47, subject to the Cambridge Core terms of use. 326 F. M. RAGAB where m is a positive integer, R(± mn +1)>0 and R(p) > 0; x is taken to be rea and positive and the asterisk has the same meaning as before.
    [Show full text]
  • Non-Classical Integrals of Bessel Functions
    J. Austral. Math. Soc. (Series E) 22 (1981), 368-378 NON-CLASSICAL INTEGRALS OF BESSEL FUNCTIONS S. N. STUART (Received 14 August 1979) (Revised 20 June 1980) Abstract Certain definite integrals involving spherical Bessel functions are treated by relating them to Fourier integrals of the point multipoles of potential theory. The main result (apparently new) concerns where /,, l2 and N are non-negative integers, and r, and r2 are real; it is interpreted as a generalized function derived by differential operations from the delta function 2 fi(r, — r^. An ancillary theorem is presented which expresses the gradient V "y/m(V) of a spherical harmonic function g(r)YLM(U) in a form that separates angular and radial variables. A simple means of translating such a function is also derived. 1. Introduction Some of the best-known differential equations of mathematical physics lead to Fourier integrals that involve Bessel functions when, for instance, they are solved under conditions of rotational symmetry about a centre. As long as the integrals are absolutely convergent they are amenable to the classical analysis of Watson's standard treatise, but a calculus of wider serviceability can be expected from the theory of generalized functions. This paper identifies a class of Bessel-function integrals that are Fourier integrals of derivatives of the Dirac delta function in three dimensions. They arise in the practical context of manipulating special functions-notably when changing the origin of the polar coordinates, for example, in order to evaluate so-called two-centre integrals and ©Copyright Australian Mathematical Society 1981 368 Downloaded from https://www.cambridge.org/core.
    [Show full text]
  • Radial Functions and the Fourier Transform
    Radial functions and the Fourier transform Notes for Math 583A, Fall 2008 December 6, 2008 1 Area of a sphere The volume in n dimensions is n n−1 n−1 vol = d x = dx1 ··· dxn = r dr d ω. (1) Here r = |x| is the radius, and ω = x/r it a radial unit vector. Also dn−1ω denotes the angular integral. For instance, when n = 2 it is dθ for 0 ≤ θ ≤ 2π, while for n = 3 it is sin(θ) dθ dφ for 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. The radial component of the volume gives the area of the sphere. The radial directional derivative along the unit vector ω = x/r may be denoted 1 ∂ ∂ ∂ ωd = (x1 + ··· + xn ) = . (2) r ∂x1 ∂xn ∂r The corresponding spherical area is ωdcvol = rn−1 dn−1ω. (3) Thus when n = 2 it is (1/r)(x dy−y dx) = r dθ, while for n = 3 it is (1/r)(x dy dz+ y dz dx + x dx dy) = r2 sin(θ) dθ dφ. The divergence theorem for the ball Br of radius r is thus Z Z div v dnx = v · ω rn−1dn−1ω. (4) Br Sr Notice that if one takes v = x, then div x = n, while x · ω = r. This shows that n n times the volume of the ball is r times the surface areaR of the sphere. ∞ z −t dt Recall that the Gamma function is defined by Γ(z) = 0 t e t . It is easy to show that Γ(z + 1) = zΓ(z).
    [Show full text]
  • Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions
    Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions Banu Yılmaz YAS¸AR and Mehmet Ali OZARSLAN¨ Eastern Mediterranean University, Department of Mathematics Gazimagusa, TRNC, Mersin 10, Turkey E-mail: [email protected]; [email protected] Abstract In the present paper, unification of Bessel, modified Bessel, spherical Bessel and Bessel- Clifford functions via the generalized Pochhammer symbol [ Srivastava HM, C¸etinkaya A, Kıymaz O. A certain generalized Pochhammer symbol and its applications to hypergeometric functions. Applied Mathematics and Computation, 2014, 226 : 484-491] is defined. Several potentially useful properties of the unified family such as generating function, integral representation, Laplace transform and Mellin transform are obtained. Besides, the unified Bessel, modified Bessel, spherical Bessel and Bessel- Clifford functions are given as a series of Bessel functions. Furthermore, the derivatives, recurrence relations and partial differential equation of the so-called unified family are found. Moreover, the Mellin transform of the products of the unified Bessel functions are obtained. Besides, a three-fold integral representation is given for unified Bessel function. Some of the results which are obtained in this paper are new and some of them coincide with the known results in special cases. Key words Generalized Pochhammer symbol; generalized Bessel function; generalized spherical Bessel function; generalized Bessel-Clifford function 2010 MR Subject Classification 33C10; 33C05; 44A10 1 Introduction Bessel function first arises in the investigation of a physical problem in Daniel Bernoulli's analysis of the small oscillations of a uniform heavy flexible chain [26, 37]. It appears when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates.
    [Show full text]
  • Special Functions
    Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS table of content Chapter 5 Special Functions 5.1 Heaviside step function - filter function 5.2 Dirac delta function - modeling of impulse processes 5.3 Sine integral function - Gibbs phenomena 5.4 Error function 5.5 Gamma function 5.6 Bessel functions 1. Bessel equation of order ν (BE) 2. Singular points. Frobenius method 3. Indicial equation 4. First solution – Bessel function of the 1st kind 5. Second solution – Bessel function of the 2nd kind. General solution of Bessel equation 6. Bessel functions of half orders – spherical Bessel functions 7. Bessel function of the complex variable – Bessel function of the 3rd kind (Hankel functions) 8. Properties of Bessel functions: - oscillations - identities - differentiation - integration - addition theorem 9. Generating functions 10. Modified Bessel equation (MBE) - modified Bessel functions of the 1st and the 2nd kind 11. Equations solvable in terms of Bessel functions - Airy equation, Airy functions 12. Orthogonality of Bessel functions - self-adjoint form of Bessel equation - orthogonal sets in circular domain - orthogonal sets in annular fomain - Fourier-Bessel series 5.7 Legendre Functions 1. Legendre equation 2. Solution of Legendre equation – Legendre polynomials 3. Recurrence and Rodrigues’ formulae 4. Orthogonality of Legendre polynomials 5. Fourier-Legendre series 6. Integral transform 5.8 Exercises Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS Introduction In this chapter we summarize information about several functions which are widely used for mathematical modeling in engineering. Some of them play a supplemental role, while the others, such as the Bessel and Legendre functions, are of primary importance. These functions appear as solutions of boundary value problems in physics and engineering.
    [Show full text]
  • Bessel Functions and Their Applications
    Bessel Functions and Their Applications Jennifer Niedziela∗ University of Tennessee - Knoxville (Dated: October 29, 2008) Bessel functions are a series of solutions to a second order differential equation that arise in many diverse situations. This paper derives the Bessel functions through use of a series solution to a differential equation, develops the different kinds of Bessel functions, and explores the topic of zeroes. Finally, Bessel functions are found to be the solution to the Schroedinger equation in a situation with cylindrical symmetry. 1 INTRODUCTION 0 0 X 2 n+s−1 (xy ) = an(n + s) x n=0 While special types of what would later be known as 1 0 0 X 2 n+s Bessel functions were studied by Euler, Lagrange, and (xy ) = an(n + s) x the Bernoullis, the Bessel functions were first used by F. n=0 W. Bessel to describe three body motion, with the Bessel functions appearing in the series expansion on planetary When the coefficients of the powers of x are organized, we perturbation [1]. This paper presents the Bessel func- find that the coefficient on xs gives the indicial equation tions as arising from the solution of a differential equa- s2 − ν2 = 0; =) s = ±ν, and we develop the general tion; an equation which appears frequently in applica- formula for the coefficient on the xs+n term: tions and solutions to physical situations [2] [3]. Fre- quently, the key to solving such problems is to recognize an−2 the form of this equation, thus allowing employment of an = − (3) (n + s)2 − ν2 the Bessel functions as solutions.
    [Show full text]
  • Circular Waveguides Circular Waveguides Introduction
    Circular Waveguides Circular waveguides Introduction Waveguides can be simply described as metal pipes. Depending on their cross section there are rectangular waveguides (described in separate tutorial) and circular waveguides, which cross section is simply a circle. This tutorial is dedicated to basic properties of circular waveguides. For properties visualisation the electromagnetic simulations in QuickWave software are used. All examples used here were prepared in free CAD QW-Modeller for QuickWave and the models preparation procedure is described in separate documents. All examples considered herein are included in the QW-Modeller and QuickWave STUDENT Release installation as both, QW-Modeller and QW-Editor projects. Table of Contents CUTOFF FREQUENCY ........................................................................................................................................ 2 PROPAGATION MODES IN CIRCULAR WAVEGUIDE ........................................................................................... 5 MODE TE11........................................................................................................................................................ 5 MODE TM01 ...................................................................................................................................................... 9 1 www.qwed.eu Circular Waveguides Cutoff frequency Similarly as in the case of rectangular waveguides, propagation in circular waveguides is determined by a cutoff frequency. The cutoff frequency
    [Show full text]
  • The Laplace Transform of the Bessel Functions <Inlineequation ID="IE1
    The Laplace T,'ansform of the Bessel Functions where n ~ 1, 2, 3~ ... by F. M. RAhaB (a Cairo, Egitto - U.A.R.) (x) oo Summary. - The integrals / e--PtI(~ (1)xt~-n tl~--~dt and f e--pttk--iJ f 2a~t=~ni 1] di, n=l, 2, 3, .. o o are evaluated in term of MacRoberts' E--Functious and iu terms of generalized hyper. geometric functions. By talcing k~l, the Laplace transform of the four functions in the title is obtained. § 1. Introduction. - Nothing is known in the literature ([1] and [2]) of the Laplace transform of the functions where n is any posit.ire integer, except the case where the argument of the Bessel functions is equal to [~cti/2] (see [1] p. 185 and p. 199). In this paper the following four formulae, which give the Laplace transform of the above functions, will be established: / co 1 (1) e-Pttk--iKp xt dt=p--k2~--~(2u)~-'~x ~ o ](2n)2"p2]~_k nk ~ h n; : e :ei~ [~4x-~--]2 E A n; ~ + 2 , 2 2: (2n)~"p ~ 1 [[27~2n~2~ i . 1 E ~ n; 2 ~- , A n; 2 : 2: (2n)2"p 2 e:~ where R(k + U) ~ o, R{pI ~ o and n is any positive integer. The symbol h(n; a) represents the set of 318 F.M. RAGAB: The Laptace Tra.ns]orm o~ the Bessel Functions, etc. parameters o~ a+l a+n--1 8 (2~ e-ptt~-I xt dt = 2 k-~ r: o [ 1 1 1 ( 1 ) ( ~, 1 I, k, k+ h n; ~ h n ~, -~i ~ 2 )' ~ , ; --~.
    [Show full text]
  • Chapter on Diffraction
    Chapter 3 Diffraction While the particle description of Chapter 1 works amazingly well to understand how ra- diation is generated and transported in (and out of) astrophysical objects, it falls short when we investigate how we detect this radiation with our telescopes. In particular the diffraction limit of a telescope is the limit in spatial resolution a telescope of a given size can achieve because of the wave nature of light. Diffraction gives rise to a “Point Spread Function”, meaning that a point source on the sky will appear as a blob of a certain size on your image plane. The size of this blob limits your spatial resolution. The only way to improve this is to go to a larger telescope. Since the theory of diffraction is such a fundamental part of the theory of telescopes, and since it gives a good demonstration of the wave-like nature of light, this chapter is devoted to a relatively detailed ab-initio study of diffraction and the derivation of the point spread function. Literature: Born & Wolf, “Principles of Optics”, 1959, 2002, Press Syndicate of the Univeristy of Cambridge, ISBN 0521642221. A classic. Steward, “Fourier Optics: An Introduction”, 2nd Edition, 2004, Dover Publications, ISBN 0486435040 Goodman, “Introduction to Fourier Optics”, 3rd Edition, 2005, Roberts & Company Publishers, ISBN 0974707724 3.1 Fresnel diffraction Let us construct a simple “Camera Obscura” type of telescope, as shown in Fig. 3.1. We have a point-source at location “a”, and we point our telescope exactly at that source. Our telescope consists simply of a screen with a circular hole called the aperture or alternatively the pupil.
    [Show full text]
  • Bessel Functions and Equations of Mathematical Physics Markel Epelde García
    Bessel Functions and Equations of Mathematical Physics Final Degree Dissertation Degree in Mathematics Markel Epelde García Supervisor: Judith Rivas Ulloa Leioa, 25 June 2015 Contents Preface 3 1 Preliminaries 5 1.1 The Gamma function . .6 1.2 Sturm-Liouville problems . .7 1.3 Resolution by separation of variables . 10 1.3.1 The Laplacian in polar coordinates . 11 1.3.2 The wave equation . 12 1.3.3 The heat equation . 13 1.3.4 The Dirichlet problem . 14 2 Bessel functions and associated equations 15 2.1 The series solution of Bessel’s equation . 15 2.2 Bessel functions of the second and third kind . 20 2.3 Modified Bessel functions . 25 2.4 Exercises . 29 3 Properties of Bessel functions 33 3.1 Recurrence relations . 33 3.2 Bessel coefficients . 37 3.3 Bessel’s integral formulas . 38 3.4 Asymptotics of Bessel functions . 40 3.5 Zeros of Bessel functions . 42 3.6 Orthogonal sets . 47 3.7 Exercises . 54 4 Applications of Bessel functions 57 4.1 Vibrations of a circular membrane . 58 4.2 The heat equation in polar coordinates . 61 4.3 The Dirichlet problem in a cylinder . 64 Bibliography 67 1 2 Preface The aim of this dissertation is to introduce Bessel functions to the reader, as well as studying some of their properties. Moreover, the final goal of this document is to present the most well- known applications of Bessel functions in physics. In the first chapter, we present some of the concepts needed in the rest of the dissertation. We give definitions and properties of the Gamma function, which will be used in the defini- tion of Bessel functions.
    [Show full text]